On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method
Tiexiang Li Tsung-Ming Huang Wen-Wei Lin Jenn-Nan Wang
Inverse Problems & Imaging 2018, 12(4): 1033-1054 doi: 10.3934/ipi.2018043

In this paper, we consider the two-dimensional Maxwell's equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of $O(1)$ and the other half of eigenvalues are positive with order of $O(10^2)$. In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

keywords: Two-dimensional transmission eigenvalue problem pseudo-chiral model transverse magnetic mode linear sampling method singular value decomposition
Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
Victor Isakov Jenn-Nan Wang
Inverse Problems & Imaging 2014, 8(4): 1139-1150 doi: 10.3934/ipi.2014.8.1139
We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schrödinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when energy level/frequency is growing. These bounds hold under certain a-priori regularity constraints on the unknown coefficient. Proofs use complex and bounded complex geometrical optics solutions.
keywords: perturbation theories. Schrödinger operator fundamental solutions quantum mechanics Inverse problems
Unique continuation property for the elasticity with general residual stress
Gunther Uhlmann Jenn-Nan Wang
Inverse Problems & Imaging 2009, 3(2): 309-317 doi: 10.3934/ipi.2009.3.309
We prove the unique continuation property for the isotropic elasticity system with arbitrarily large residual stress. This work improves the result obtained in [10] where the residual stress is assumed to be small.
keywords: Unique continuation property residual stress.
Reconstruction of obstacles immersed in an incompressible fluid
Horst Heck Gunther Uhlmann Jenn-Nan Wang
Inverse Problems & Imaging 2007, 1(1): 63-76 doi: 10.3934/ipi.2007.1.63
We consider the reconstruction of obstacles inside a bounded domain filled with an incompressible fluid. Our method relies on special complex geometrical optics solutions for the stationary Stokes equation with a variable viscosity.
keywords: inverse problem. complex spherical waves Stokes system
Optimal three-ball inequalities and quantitative uniqueness for the Stokes system
Ching-Lung Lin Gunther Uhlmann Jenn-Nan Wang
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1273-1290 doi: 10.3934/dcds.2010.28.1273
We study the local behavior of a solution to the Stokes system with singular coefficients in $R^n$ with $n=2,3$. One of our main results is a bound on the vanishing order of a nontrivial solution $u$ satisfying the Stokes system, which is a quantitative version of the strong unique continuation property for $u$. Different from the previous known results, our strong unique continuation result only involves the velocity field $u$. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-ball inequalities for $u$. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution $u$ to the Stokes system from those three-ball inequalities. As an application, we derive a minimal decaying rate at infinity of any nontrivial $u$ satisfying the Stokes equation under some a priori assumptions.
keywords: Optimal three-ball inequalities Carleman estimates Stokes system.

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